Considerations

  1. Power for TTE related to several choices:

    • Design (Parallel, One-Arm, Complex): This refers to the type of clinical trial design chosen.
      • Parallel design involves multiple groups (e.g., a treatment group vs. a control group) being studied simultaneously.
      • One-Arm design involves only one group, usually receiving the treatment or intervention.
      • Complex design might involve multiple treatments, crossover between treatments, or multiple phases.
    • Statistical Method (Log-Rank, Cox): These are methods used to analyze TTE data.
      • Log-Rank test is used to compare the survival distributions of two groups.
      • Cox proportional hazards model is a regression model used to examine the effect of several variables on survival time simultaneously.
    • Endpoint (Hazard Ratio, Survival Time):
      • Hazard Ratio (HR) measures the effect of an intervention on the hazard or risk of an event occurring at any given point in time.
      • Survival Time refers to the time until an event (like death, disease progression) occurs.
    • Survival Distribution (Exponential, Weibull): These are statistical distributions used to model survival times.
      • Exponential distribution assumes a constant hazard rate over time.
      • Weibull distribution can model varying hazard rates over time.

  2. Power driven primarily by number of events (E) not sample size (N):

    • In TTE analysis, the power to detect a difference or effect is more influenced by the number of events (e.g., deaths, disease recurrences) rather than just the number of participants. This is because the statistical significance in such analyses often depends more on the event occurrence across the follow-up period than the sheer number of participants.

  3. Calculating E separate from N:

  • E (number of events needed to achieve adequate power) is often determined separately from N (number of participants). This allows for a more nuanced understanding of what is required statistically to observe a meaningful difference or effect.
  • Recruitment Strategy: How patients are enrolled over time, whether accrual is expected to be uniform or variable (accelerating towards the end or starting strong and tapering), impacts both the study timeline and its feasibility.
  • Study Length Considerations:
    • Event-Driven: Waiting to reach a predetermined number of events before concluding the study, which can ensure sufficient data for robust conclusions but may prolong the study duration.
    • Time-Driven: Fixed study durations based on predefined time points, which can streamline operations and planning but might result in underpowered results if insufficient events have occurred.

  1. Accrual/Follow-up

    • Follow-up/Study Length: The length of follow-up for each subject, which can be fixed or until a certain number of events occur, influences the censored data and overall study duration. The strategy chosen impacts how data censoring is handled, affecting statistical power and analysis.
    • Accrual: This refers to how participants are enrolled over time—whether accrual happens at a uniform rate or follows a more complex pattern like a truncated exponential. The accrual model affects the pace at which data is collected and the feasibility of study timelines.
    • Dropout: Addressing how dropout is modeled—either as a simple percentage or via more sophisticated survival-like models that estimate dropout as a function of time or hazard.
    • For complex trial designs, total follow-up time (the time period during which participants are observed) becomes critical because it impacts the number of events that can be observed.

  2. Survival Distribution/Effect Size

    • Sample size calculations in TTE studies are focused on determining the number of subjects needed to observe a sufficient number of events to achieve statistical power.
    • Survival/Distribution: This includes parameters like the hazard rate, which might be constant or vary over time (piecewise), and the median survival time. Survival distributions such as exponential or Weibull are parametric forms used to estimate these characteristics.
    • Effect Size Choice & Estimate: Decisions on what effect size to measure, such as the hazard ratio, relative time differences between treatments, or survival time differences. These are crucial for defining the clinical relevance and statistical detectability of the trial outcomes.

  3. Other Consideration:

  • A flexible meta-model is used to incorporate different aspects of a clinical trial, such as rate of participant accrual (how quickly participants are enrolled), dropout rates (how many leave the study before completion), and crossover (participants switching from one treatment group to another). This model helps in estimating both E and N realistically.
  • Use of Progression-Free Survival (PFS) for Accelerated Approval
    • Context: In trials for rare diseases or conditions where quicker approvals are desirable, regulatory agencies might allow accelerated approval based on interim endpoints like PFS. This strategy enables faster access to treatments that show promise without waiting for final OS (Overall Survival) data.
    • Application: PFS as an endpoint can expedite drug approval processes, allowing for earlier patient access while continuing to monitor long-term benefits such as OS in a comprehensive manner.
  • Choice of Test Statistics: Hazard Ratios vs. Other Metrics
    • Hazard Ratios: Commonly used due to their effectiveness in comparing the risk of an event between two groups over time. However, hazard ratios can be complex to interpret, especially in communicating how they translate into clinical benefit.
    • Alternative Metrics: Restricted Mean Survival Time (RMST) or other point estimates like survival proportions at specific times can be easier to communicate and may provide more directly interpretable clinical relevance.

Log Rank Test (Freedman)

Reference

Note:

  1. Sample size and power of the logrank test to be analyzed under the assumption of proportional hazards.
  2. Time periods are not stated (without impact of accrual and follow-up time). Rather, it is assumed that enough time elapses to allow for a reasonable proportion of responses to occur.
  3. The formulas used in this module come from Machin et al. (2018). They are also given in Fayers and Machin (2016) where they are applied to sizing quality of life studies. They were originally published in Freedman (1982) and are often referred to by that name. Freedman, L.S. 1982. ‘Tables of the Number of Patients Required in Clinical Trials using the Logrank Test’.Statistics in Medicine, Vol. 1, Pages 121-129.
  4. The power calculations used here assume proportional hazards and are based on the number of events.
  5. In order to estimate sample sizes, an additional assumption is made that the underlying exponential distribution. First, the logrank test and the test derived using the exponential distribution have nearly the same power when the data are in fact exponentially distributed. Second, under the proportional hazards model (which is assumed by the logrank test), the survival distribution can be transformed to be exponential and the logrank test remains the same under monotonic transformations.

We assume that a study is to be made comparing the survival (or healing) of a control group with an experimental group. The control group (group 1) consists of patients that will receive the existing treatment. In cases where no existing treatment exists, the group 1 consists of patients that will receive a placebo. The experimental group (group 2) will receive the new treatment. We assume that the critical event of interest is death and that two treatments have survival distributions with instantaneous death (hazard) rates, 𝜆1 and 𝜆2. These hazard rates are a subject’s probability of death in a short period of time.

There are several ways to compare two hazard rates. One is the difference, \(\lambda_2-\lambda_1\). Another is the ratio, \(\lambda_2 / \lambda_1\), called the hazard ratio. \[ H R=\frac{\lambda_2}{\lambda_1} \] Note that since HR is formed by dividing the hazard rate of the experimental group by that of the control group, a treatment that has a smaller hazard rate than the control will have a hazard ratio that is less than one.

The hazard ratio may be formulated in other ways. If the proportions surviving during the study are called \(S 1\) and \(S 2\) for the control and experimental groups, the hazard ratio is given by \[ H R=\frac{\log \left(S_2\right)}{\log \left(S_1\right)} \] Furthermore, if the median survival times of the two groups are \(M 1\) and \(M 2\), the hazard ratio is given by \[ H R=\frac{M_1}{M_2} \]

We assume that the logrank test will be used to analyze the data once they are collected. However, often Cox’s proportional hazards regression is used to do the actual analysis. The power calculations of the logrank test are based on several other parameters \[ z_{1-\beta}=\frac{|H R-1| \sqrt{N(1-w) \varphi\left[\left(1-S_1\right)+\varphi\left(1-S_2\right)\right] /(1+\varphi)}}{(1+\varphi H R)}-z_{1-\alpha / k} \] where \(k\) is 1 for a one-sided hypothesis test or 2 for a two-sided test, \(\alpha\) and \(\beta\) are the error rates defined as usual, the \(z^{\prime}\) s are the usual points from the standard normal distribution, \(w\) is the proportion that are lost to follow up, and \(\varphi\) represents the sample size ratio between the two groups. \[ \varphi=\frac{N_2}{N_1} \] Note that the null hypothesis is that the hazard ratio is one, i.e., that \[ H_0: \frac{\lambda_2}{\lambda_1}=1 \]

Calculation in R (Survival Rate)

ssc.logRank.Freedman <- function(S.trt, S.ctrl, sig.level = 0.05, power = 0.8, 
                        alternative = c("two.sided", "less", "greater"),
                        method = c("Freedman"),
                        pr=TRUE) {
  
  # FIXME: Relabel S.trt and S.ctrl as S.ctrl and S.trt
  alt <- match.arg(alternative)
  za <- if (alt == "two.sided") {
    stats::qnorm(sig.level / 2) 
  } else {
    stats::qnorm(sig.level)
  }
  zb <- stats::qnorm(1 - power)
  haz.ratio <- log(S.trt) / log(S.ctrl)
  if(pr)
    cat("\nHazard ratio:",format(haz.ratio),"\n")
  
  cat("Expected number of events:", 4 * (za + zb) ^ 2 / log(1 / haz.ratio) ^ 2)
  cat("\n")
  (((haz.ratio + 1) / (haz.ratio - 1)) ^ 2) * 
    (za + zb) ^ 2 / (2 - S.trt - S.ctrl)
}
ssc.logRank.Freedman(0.5,0.7,power = 0.817) 
## 
## Hazard ratio: 1.943358 
## Expected number of events: 74.32079
## [1] 99.81032
## HR
## log(0.7) / log(0.5)

Calculation in R (Median Survival)

Using an unstratified log-rank test at the one-sided 2.5% significance level, a total of 282 events would allow 92.6% power to demonstrate a 33% risk reduction (hazard ratio for RAD/placebo of about 0.67, as calculated from an anticipated 50% increase in median PFS, from 6 months in placebo arm to 9 months in the RAD001 arm).

With a uniform accrual of approximately 23 patients per month over 74 weeks and a minimum follow up of 39 weeks, a total of 352 patients would be required to obtain 282 PFS events, assuming an exponential progression-free survival distribution with a median of 6 months in the Placebo arm and of 9 months in RAD001 arm. With an estimated 10% lost to follow up patients, a total sample size of 392 patients should be randomized.

Yao JC, Shah MH, Ito T, Bohas CL, Wolin EM, Van Cutsem E, Hobday TJ, Okusaka T, Capdevila J, de Vries EG, Tomassetti P, Pavel ME, Hoosen S, Haas T, Lincy J, Lebwohl D, Öberg K; RAD001 in Advanced Neuroendocrine Tumors, Third Trial (RADIANT-3) Study Group. Everolimus for advanced pancreatic neuroendocrine tumors. N Engl J Med. 2011 Feb 10;364(6):514-23. doi: 10.1056/NEJMoa1009290. PMID: 21306238; PMCID: PMC4208619. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4208619/

Significance Level (1-Sided) 0.025
Placebo Median Survival (months) 6
Everolimus Median Survival (months) 9
Hazard Ratio 0.66667
Accrual Period (Weeks) 74
Minimum Follow-Up (Weeks) 39
Power % (under constant HR) 92.6 
# Load required library
library(powerSurvEpi)

# Define parameters
power <- 0.926
alpha <- 0.025  # One-sided significance level
k <- 74 / 52  # Accrual time in years
followup_time <- 39 / 52  # Minimum follow-up time in years
median_survival_placebo <- 6 / 12  # Median survival in years for placebo
median_survival_treatment <- 9 / 12  # Median survival in years for treatment
dropout_rate <- 0.10

# Calculate event rates assuming exponential survival
# Event Rates (pC, pE)**: These are derived from the median survival times assuming an exponential survival model.
pC <- log(2) / median_survival_placebo  # Event rate for placebo
pE <- log(2) / median_survival_treatment  # Event rate for treatment
RR <- 0.66667  # Risk reduction as hazard ratio

# Calculate the required sample size
sample_size <- ssizeCT.default(power = power, 
                               k = k + followup_time, # Total study duration including follow-up
                               pE = pE, 
                               pC = pC, 
                               RR = RR, 
                               alpha = alpha)

# Adjust for dropout
final_sample_size <- sample_size / (1 - dropout_rate)
final_sample_size <- ceiling(final_sample_size)  # Round up to the next whole number

# Output the calculated sample size
print(paste("Total sample size needed, accounting for dropout:", final_sample_size))
## [1] "Total sample size needed, accounting for dropout: 242"
## [2] "Total sample size needed, accounting for dropout: 112"

Calculation in SAS (Median Survival)

proc power;
   twosamplelogrank
      test=logrank
      groupmedians = (6 9)   /* Median survival times in months */
      hazardratio = 0.66667  /* Hazard Ratio */
      accrualtime = 74/52     /* Accrual time in years */
      followuptime = 39/52    /* Follow-up time in years */
      sides = 1               /* One-sided test */
      power = 0.926           /* Desired power */
      alpha = 0.025           /* One-sided significance level */
      ntotal = .              /* Let SAS calculate required total sample size */
      groupweights = (1 1);   /* Equal weighting of groups */
run;

Log Rank Test

Reference

Introduction

One reason of log-rank tests are useful is that they provide an objective criteria (statistical significance) around which to plan out a study:

  1. How many subjects do we need?
  2. How long will the study take to complete?

In survival analysis, we need to specify information regarding the censoring mechanism and the particular survival distributions in the null and alternative hypotheses.

  • First, one needs either to specify what parametric survival model to use, or that the test will be semi-parametric, e.g., the log-rank test. This allows for determining the number of deaths (or events) required to meet the power and other design specifications.
  • Second, one must also provide an estimate of the number of patients that need to be entered into the trial to produce the required number of deaths.

We shall assume that the patients enter a trial over a certain accrual period of length \(a\), and then followed for an additional period of time \(f\) known as the follow-up time. Patients still alive at the end of follow-up are censored.

Exponential Approximation

In general, it is assumed we have constant hazards (i.e., exponential distributions) for the sake of simplicity. Because other work in literature has indicated that the power/sample size obtained from assuming constant hazards is fairly close to the empirical power of the log-rank test, provided that the ratio between the two hazard functions is constant. Typically in a power analysis, we are simply trying to find the approximate number of subjects required by the study, and many approximations/guesses are involved, so using formulas based on the exponential distribution is often good enough.

Calculation in R

Reference

  • Peterson B, George SL: Controlled Clinical Trials 14:511–522; 1993.
  • Lachin JM, Foulkes MA: Biometrics 42:507–519; 1986.
  • Schoenfeld D: Biometrics 39:499–503; 1983.

Assumes exponential distributions for both treatment groups. Uses the George-Desu method along with formulas of Schoenfeld that allow estimation of the expected number of events in the two groups. To allow for drop-ins (noncompliance to control therapy, crossover to intervention) and noncompliance of the intervention, the method of Lachin and Foulkes is used.

For handling noncompliance, uses a modification of formula (5.4) of Lachin and Foulkes. Their method is based on a test for the difference in two hazard rates, whereas cpower is based on testing the difference in two log hazards. It is assumed here that the same correction factor can be approximately applied to the log hazard ratio as Lachin and Foulkes applied to the hazard difference.

Note that Schoenfeld approximates the variance of the log hazard ratio by 4/m, where m is the total number of events, whereas the George-Desu method uses the slightly better 1/m1 + 1/m2. Power from this function will thus differ slightly from that obtained with the SAS samsizc program.

## 
## Accrual duration: 1.5 years  Minimum follow-up: 5 years
## 
## Total sample size: 950 
## 
## Alpha= 0.05 
## 
## 5-year Mortalities (Events Rate)
##      Control Intervention 
##         0.18         0.10 
## 
## Hazard Rates
##      Control Intervention 
##   0.03969019   0.02107210 
## 
## Probabilities of an Event During Study
##      Control Intervention 
##    0.2039322    0.1140750 
## 
## Expected Number of Events
##      Control Intervention 
##         96.9         54.2 
## 
## Hazard ratio: 0.5309147 
## 
## Drop-in rate (controls):10%
## Non-adherence rate (intervention):15%
## Effective hazard ratio with non-compliance: 0.6219687 
## Standard deviation of log hazard ratio: 0.1696421 
## Approximation method of variance of the log hazard ratio based on Peterson B, George SL: Controlled Clinical Trials 14:511–522; 1993. 
## 
##     Power 
## 0.7993381
## 
## Accrual duration: 1.5 years  Minimum follow-up: 5 years
## 
## Total sample size: 950 
## 
## Alpha= 0.05 
## 
## 5-year Mortalities (Events Rate)
##      Control Intervention 
##         0.18         0.10 
## 
## Hazard Rates
##      Control Intervention 
##   0.03969019   0.02107210 
## 
## Probabilities of an Event During Study
##      Control Intervention 
##    0.2039322    0.1140750 
## 
## Expected Number of Events
##      Control Intervention 
##         91.8         57.0 
## 
## Hazard ratio: 0.5309147 
## 
## Drop-in rate (controls):10%
## Non-adherence rate (intervention):15%
## Effective hazard ratio with non-compliance: 0.6219687 
## Standard deviation of log hazard ratio: 0.1639526 
## Approximation method of variance of the log hazard ratio based on Schoenfeld D: Biometrics 39:499–503; 1983. 
## 
##     Power 
## 0.8254654

Calculation in SAS

Scenario 1

Patients will be accrued uniformly over two years and then followed for an additional three years past the accrual period. Some loss to follow-up is expected, with roughly exponential rates that would result in about 50% loss with the standard treatment within 10 years. The loss to follow-up with the proposed treatment is more difficult to predict, but 50% loss would be expected to occur sometime between years 5 and 20.

## time at event estimated = 2
## duration of accrual period = 2
## minimum follow-up time = 3
## Standard treatment: 50% loss with the standard treatment within 10 years
## Proposed treatment: 50% loss would be expected to occur sometime between years 5 and 20
## The "Standard" curve specifying an exponential form with a survival probability of 0.5 at year 5.
## The "Proposed" curve is a piecewise linear curve defined by the five points shown

proc power;
    twosamplesurvival test= logrank 
    accrualtime=2
    followuptime=3
    power = 0.8
    alpha = 0.05
    sides = 2 
    curve("Standard") = 5 : 0.5
    curve("Proposed") = (1 to 5 by 1):(0.95 0.9 0.75 0.7 0.6)
    groupsurvival = "Standard" | "Proposed"
    groupmedlosstimes = 10 | 20 5 
    npergroup = .;  
run;

Scenario 2

  1. 30% of placebo patients are sustained responders (exponential hazard =0.3567)
  2. 45 or 50% for the treatment group (exp. hazard = 0.5978 or 0.6931)
  3. Twice as many patients are on treatment as placebo 1:2 Groupweights statement 2:1 randomization ratio
  4. All patients are enrolled at the beginning of the study with a 30% drop-out rate The dropout rate were also converted to group loss hazards in the same way. Therefore, the 30% dropout rate was corresponding to the group loss hazard of -ln(1-dropout rate)=-ln(1-0.3)=0.3567.
data __NULL_;
    HR1 = -log(1-0.3);
    HR2a = -log(1-0.45);
    HR2b = -log(1-0.5);
    put HR1 HR2a HR2b;
run;
proc power;
    twosamplesurvival test=logrank
    /* Specify Analysis Information */
    accrualtime=2
    followuptime=3
    power = 0.8
    alpha = 0.05
    sides = 2
    /* Specify Effects */
    gexphs= 0.3567 | 0.5978 .6931
    groupweights = (2 1)
    /* Specify Loss Information */
    grouplossexphazards=(0.3567 0.3567)
    ntotal= .;
    plot y=power min=0.5 max=0.90;
run;

Scenario 3

Clinical trial to assess new treatment for patients with chronic active hepatitis.

  1. Under standard treatment, 41% of patients survive beyond 5 years
  2. Expect new treatment to increase survival beyond 5 years to 60%.

Calculation

  • Event rate for standard treatment (Ec) = 1-0.41 = 0.59
  • Event rate for new treatment (Et) = 1-0.60 = 0.4
  • Since event rate E = 1 - exp(-t*HAZARD), we have HAZARD = -ln((1-E)/t
  • The Hazard for standard treatment is HAZARDc=-ln(1-Ec)/t = -ln(1-0.59)/t = -ln(0.41)/t
  • The Hazard for new treatment is HAZARDt = -ln(1-Et)/t = -ln(1-0.40)/t = -ln(0.60)/t
  • The hazard ratio = HAZARDt/HAZARDc = ln((0.6)/ln(0.41)=0.5729
  • T=5, the hazard for standard treatment is HAZARDc = -ln(0.41)/5 = 0.178
proc power;
    twosamplesurvival test=logrank
    /* Specify Analysis Information */
    followuptime = 5
    totalTIME = 5
    power = 0.8
    alpha = 0.05
    sides = 2
    /* Specify Effects */
    hazardratio = 0.57
    refsurvexphazard=0.178
    ntotal = . ;
run;

Scenario 4

  • Accrual time is accepted as half of the total study time, which is equal to 5 years.
  • Changing curve parameter to curve(“Control”) = (5):(0.8)will give the exponential distribution of the event
proc power;
     twosamplesurvival
     test=logrank
     curve("Control") = (0 5):(1 0.8)
     curve("Treatment") = (0 5):(1 0.85)
     refsurvival = "Control"
     accrualtime = 2.5
     followuptime = 2.5 
     hazardratio = 1.373
     alpha = 0.05
     sides = 2
     ntotal = .
     power = 0.8;
 run;

Scenario 5

Piecewise linear survival curve

  1. The survival curve of patients for the existing treatment is known to be approximately exponential with a median survival time of 5 years
  2. proposed treatment will yield a survival curve described by the times and probabilities
    • Time 1 0.95
    • Time 2 0.90
    • Time 3 0.75
    • Time 4 0.70
    • Time 5 0.60
proc power;
      twosamplesurvival test=logrank
       curve("Existing Treatment") = 5 : 0.5
      curve("Proposed Treatment") = 1 : 0.95 2 : 0.90 3:0.75  4:0.70 5:0.60
      groupsurvival = "Existing Treatment" | "Proposed Treatment"
      accrualtime = 2
      FOLLOWUPTIME = 3
      power = 0.80
      alpha=0.05
      npergroup = . ;
run;

Scenario 6

Group sequential design with interim analyses

the survival probability at 12 months are for standard and proposed groups are specified the statement of grouplossexphazards is used to account for the dropout rate.

proc power;
      twosamplesurvival test=logrank
      curve("Standard") = 12 : 0.8781
      curve("Proposed") = 12 : 0.9012
      groupsurvival = "Standard" | "Proposed"
      accrualtime = 18
      Totaltime = 24
      GROUPLOSSEXPHAZARDS = (0.0012 0.0012)
      NSUBINTERVAL = 1
      power = 0.85
      ntotal = . ;
run;

Log Rank Test with Competing Risk

Reference

This procedure is based on the formulas presented in Pintilie (2006) and Machin et al. (2009), which are both based on the original paper Pintilie (2002).

  • Logrank Tests Accounting for Competing Risks
  • Machin, D., Campbell, M.J., Tan, S.B., Tan, S.H. 2009. Sample Size Tables for Clinical Studies, Third Edition. Wiley-Blackwell, Chichester, United Kingdom.
  • Pintilie, M., 2006. Competing Risks: A Practical Perspective. John Wiley & Sons, Chichester, United Kingdom.
  • Pintilie, M., 2002. ‘Dealing with Competing Risks: Testing Covariates and Calculating Sample Size’. Statistics in Medicine, Volume 21, pages 3317-3324.

Introduction

Logrank test is used to compare the two survival distributions because it is easy to apply and is usually more powerful than an analysis based simply on proportions. It compares survival across the whole spectrum of time, not at just one or two points, and accounts for censoring.

When analyzing time-to-event data and calculating power and sample size, a complication arises when individuals in the study die from risk factors that are not directly related to the risk factor of interest. For example, a researcher may wish to determine if a new drug for some disease improves patient survival time when compared to a standard treatment. Therefore, the researchers would be interested to know how long each patient lives until he or she dies from the disease. However, during the course of the study, patients may also die from other risks such as myocardial infarction, diabetes, or even an accident. When a patient dies from one of these other risk factors, then the main event of interest cannot be observed, so the true time-to-event of the disease for that patient can never be determined.

Power Overestimated

If the results are not adjusted, then the power calculated for the logrank test of the main event of interest may be grossly overestimated, depending on the incidence of competing risks

Assumptions

The power and sample size calculations in the module for the logrank test are based on the following assumptions:

  1. Failure times for the event of interest and competing risks are independent.
  2. Failure times are exponentially distributed.
  3. Uniform entry of subjects into the trial during the accrual period.

Details

The hazard rates for the event of interest and competing risks in group \(i\) are calculated from the cumulative survival functions as \[ \begin{aligned} & h_{e v, i}=\left(\frac{-\ln \left(S_{e v, i}(T 0)\right)}{T 0}\right) \\ & h_{c r, i}=\left(\frac{-\ln \left(S_{c r, i}(T 0)\right)}{T 0}\right) \end{aligned} \] The hazard ratio used in power calculations is calculated from the hazard rates for the event of interest as \[ H R=\left(\frac{h_{e v, 2}}{h_{e v, 1}}\right) \] the hazard rate for the treatment group divided by the hazard rate for the control group. The hazard rates may be calculated using cumulative survival proportions or cumulative incidences as described above.

Then we can calculate Probability of Event and Number of Event

Probability of Event

With the hazard rates for the event of interest and competing risks, the probability of observing the event of interest in a subject in group \(i, P r_{e v, i}\), is given as \[ P r_{e v, i}=\frac{h_{e v, i}}{h_{e v, i}+h_{c r, i}}\left(1-\frac{\exp \left\{-(T-R) \times\left(h_{e v, i}+h_{c r, i}\right)\right\}-\exp \left\{-T \times\left(h_{e v, i}+h_{c r, i}\right)\right\}}{R \times\left(h_{e v, i}+h_{c r, i}\right)}\right), \] where \(T\) is the total time of trial and \(R\) is the accrual time. The follow-up time is calculated from \(T\) and \(R\) as \[ \text { Follow-Up Time }=T-R \text {. } \] The overall probability of observing the event of interest during the study in both groups is given as \[ P r_{e v}=p_1 P r_{e v, 1}+\left(1-p_1\right) P r_{e v, 2} \] where \(p_1\) is the proportion of subjects in group 1 , the control group.

Number of Events

When dealing with time-to-event data, it is the number of events observed, not the total number of subjects that is important to achieve the specified power. The total required number of events (for the event of interest), \(E\), is calculated from the total sample size \(N\) and \(P r_{e v}\) as \[ E=N \times P r_{e v} \] The number of events in group \(i\) is calculated as \[ E_i=n_i \times P r_{e v, i} \] where \(n_i\) is the sample size for the \(i^{i \text { th }}\) group.

Power and Sample Size Calculations

Assuming an exponential model and independence of failure times for the event of interest and competing risks, Pintilie (2006) gives the following equation relating E (total number of events for the risk factor of interest) and power:

\[ z_{1-\beta}=\sqrt{E \times p_1\left(1-p_1\right)} \log (H R)-z_{1-\alpha / 2} \] with

  • \(\alpha \quad\) probability of type I error
  • \(\beta \quad\) probability of type II error
  • \(z_{1-\alpha / 2}\) standard normal quantile for \(1-\alpha / 2\)
  • \(z_{1-\beta} \quad\) standard normal quantile for \(1-\beta\)
  • \(E \quad\) total number of events for the risk factor of interest
  • \(p_1 \quad\) proportion of subjects in group 1, the control group
  • HR hazard ratio to detect

This power formula indicates that it is the total number of events observed, not the number of subjects that is critical for achieving the desired power for the logrank test.

The power formula can be rearranged to solve for \(E\), the total number of events required. The formula is \[ E=\left(\frac{1}{p_1\left(1-p_1\right)}\right) \times\left(\frac{z_{1-\alpha / 2}+z_{1-\beta}}{\log (H R)}\right)^2 . \] The overall sample size can be computed from \(E\) and \(P r_{e v}\) as \[ N=\frac{E}{P r_{e v}}=\left(\frac{1}{p_1\left(1-p_1\right) \times P r_{e v}}\right) \times\left(\frac{z_{1-\alpha / 2}+z_{1-\beta}}{\log (H R)}\right)^2 . \] The individual group sample sizes are calculated as \[ \begin{aligned} & n_1=N \times p_1, \\ & n_2=N \times\left(1-p_1\right), \end{aligned} \] where \(p_1\) is the proportion of subjects in group 1 , the control group.

Calculation in R

Alternative Hypothesis: Two-Sided
Alpha: 0.05
R (Accrual Time): 3
T-R (Follow-Up Time): 2 
T0 (Fixed Time Point): 3
Sev1(T0) (Control): 0.5
HR (Hazard Ratio = hev2 / hev1): 0.5
Scr1(T0) (Control): 0.4
Percent in Group 1: 50
Power: 0.6162274
Total Power (N): 150
## 
## Sample Size Calculation using Logrank Tests Accounting for Competing Risks
## Alpha 0.025 
## Power 61.62274 %
## 
## Accrual time of survival rate observed: 3 years
## Total time of tria: 5 years
## Follow-Up Time: 2 years
## 
## Survival probability for the event of interest in group 1: 0.5 
## Survival probability for the event of interest in group 2: 0.7071068 
## Hazard Ration: 0.5 
## 
## Competing risks probability: 0.4 
## 
## Proportion of subjects in group 1: 0.5 
## Proportion of subjects in group 2: 0.5 
## 
## The probability of observing the event of interest in a subject during the study for the group 1: 0.3574638 
## The probability of observing the event of interest in a subject during the study for the group 2: 0.2072824 
## 
## The number of events required for the group 1: 27 
## The number of events required for the group 2: 16 
## The total number of events required for the study: 43 
## 
## The sample sizes for the group 1: 75 
## The sample sizes for the group 2: 75 
## The total sample size of both groups combined: 150

With Interim Analysis

##   N1_Event IA   N2_Event IA N1_Patient IA N2_Patient IA     NTotal IA  N_Patient FU         Power 
##         62.00          3.00        134.00         15.00        149.00         32.00         91.24

Non-Proportional Hazards Methods

Method Description
Log-Rank “Average Hazard Ratio” – same as from univariate Cox Regression model
Linear-Rank (Weighted) Gehan-Breslow-Wilcoxon, Tarone-Ware, Farrington-Manning, Peto-Peto, Threshold Lag, Modestly Weighted Linear-Rank (MWLRT)
Piecewise Linear-Rank Piecewise Parametric, Weighted Piecewise Model (e.g. APPLE), Change Point Models
Combination Maximum Combination (MaxCombo) Test Procedure
Survival Time Milestone Survival (KM), Restricted Mean Survival Time, Landmark Analysis
Relative Time Ratio of Times to Reach Event Proportion, Accelerated Failure Time Models
Others Responder-Based, Frailty Models, Renyi Models, Net Benefit (Buyse)

Maximum Combination (MaxCombo) Test Overview

1. Concept: - The MaxCombo test is designed to handle multiple linear-rank tests simultaneously and to select the “best” test from the candidate tests. This approach helps in controlling Type I error rates while still allowing flexibility in the choice of statistical tests.

2. Test Variants: - Various forms of the Fleming-Harrington family of tests (denoted as F-H(G) Tests) are used, each specified by different parameterizations (G(p,q)) that emphasize different portions of the survival curve. For example, some may focus more on early failures while others on late failures.

F-H (G) Tests Proposal
G(0,1; 1,0) Lee (2007)
G(0,0*; 0,1; 1,0) Karrison (2016)
G(0,0; 0,1; 1,0; 1,1) Lin et al (2020)
G(0,0; 0,0.5; 0.5,0; 0.5,0.5) Roychoudhury et al (2021)
G(0,0; 0,0.5) Mukhopadhyay et al (2022)
G(0,0; 0,0.5; 0.5,0) Mukhopadhyay et al (2022)

3. Common Usage: - Typically, 2-4 candidate tests are considered with Fleming-Harrington being popular due to its flexibility. It can accommodate Log-Rank and Peto-Peto tests, among others, allowing researchers to tailor the analysis to the specific characteristics of their survival data.

Issues with MaxCombo Tests

1. Type I Error and Estimand: - Critics point out that MaxCombo tests, while versatile, can sometimes lead to significant results even when the treatment effect is not better than the control across all times. This can mislead the conclusions about a treatment’s efficacy, especially if it is only effective late in the follow-up period (late efficacy).

2. Interpretability: - There are concerns about the interpretability of using an average hazard ratio as the estimand because it might not accurately reflect the dynamics of the treatment effect over time, particularly under non-proportional hazards scenarios.

3. Alternatives for Improvement: - Modifications to the Fleming-Harrington weights (G(p,q) parameters) are suggested to better handle scenarios with non-proportional hazards. For example, changing the focus from early to late survival times can be achieved by adjusting these parameters.

4. Communication of Results: - It’s recommended to use the MaxCombo for analytical purposes but to communicate the results using more interpretable measures such as the Restricted Mean Survival Time (RMST), which provides a direct, clinically meaningful measure of survival benefit.

One Sample Log-Mean Method

Reference

Introduction

We will use \(\hat{\theta}\) as our test statistic, and reject \(H_0\) in favor of \(H_A\) if \(\hat{\theta}>k\) for some constant \(k\). - The significance level of the test, or Type I error rate, is \(\alpha=P\left(\hat{\theta}>k \mid \theta=\theta_0\right)\). 。 If \(Z=\frac{\hat{\theta}-\theta}{1 / \sqrt{d}}\), then we have \(\alpha=P\left(Z>\frac{k-\theta_0}{1 / \sqrt{d}}\right)\). 。 Let \(\Phi\left(z_\alpha\right)=1-\alpha\), then \(z_\alpha=\frac{k-\theta_0}{1 / \sqrt{d}}\) and hence \(k=\theta_0+\frac{z_\alpha}{\sqrt{d}}\). - The power of the test is given by \[ 1-\beta=P\left(\hat{\theta}>k \mid \theta=\theta_A\right)=P\left(Z>\frac{k-\theta_A}{1 / \sqrt{d}}\right) \] - Solving for \(d\) we have \[ \begin{gathered} z_{1-\beta}=-z_\beta=\sqrt{d}\left(k-\theta_A\right)=\sqrt{d}\left(\theta_0+\frac{z_\alpha}{\sqrt{d}}-\theta_A\right) \\ \Rightarrow d=\frac{\left(z_\beta+z_\alpha\right)^2}{\left(\theta_A+\theta_0\right)^2}=\frac{\left(z_\beta+z_\alpha\right)^2}{(\log \Delta)^2} . \end{gathered} \]

Probability of Event

Calculate patient/subject needed based on Probability of Event

We need to provide an estimate of the proportion \(\pi\) of patients who will die by the time of analysis. - If all patients entered at the same time, we would simply have \(\pi=1-S_\lambda(t)\), where \(t\) is the follow-up time. - However, patients actually enter over an accrual period of length \(a\) and then, after accrual to the trial has ended, they are followed for an additional time \(f\). - So a patient who enters at time \(t=0\) will have failure probability \(\pi(0)=1-S_\lambda(a+f)\) as this patient will have the maximum possible follow-up time \(a+f\). - Similarly, for any patient who enters at a time \(t \in[0, a]\), the failure probability \(\pi(t)=1-S_\lambda(a+f-t)\). - Assuming that the patients enter uniformly between times 0 and \(a\), the probability of death can be computed as \[ \pi=\int_0^a \frac{1}{a}\left[1-S_\lambda(a+f-t)\right] d t . \] - Assuming \(S_\lambda(t)=e^{-\lambda t}\), we have \[ \pi=1-\frac{1}{a \lambda}\left[e^{-\lambda f}-e^{-\lambda(a+f)}\right] . \]

Calculation in R (Event)

Suppose that we are designing a Phase II oncology trial where we plan a 5% level (one-sided) test, and we need 80% power to detect a hazard ratio of 1.5. We can find the required number of deaths as follows:

## Log-mean based approach
## Expected number of events
ssc.onesample.logMean(HR = 1.5, sig.level = 0.05, power = 0.8)
## 
## Hazard ratio: 1.5 
## Alpha (one-sided): 0.05 
## Power: 80 %
## 
## Log-mean based approach 
## Expected number of events: 38

Calculation in R (Patient)

We wanted to design a Phase II oncology trial where we plan a \(5 \%\) level (one-sided) test, and we need \(80 \%\) power to detect a hazard ratio of 1.5 .

Suppose that \(\lambda_0=0.15\), then we have \(\lambda_A=\lambda_0 / \Delta=0.1\). Assume accrual period \(a=2\) years and follow-up time \(f=3\) years. The probability of death under \(H_A: \lambda=0.1\) is computed as:

ssc.onesample.logMean2(HR = 1.5, sig.level = 0.05, power = 0.8, lambda=0.10, accrual=2, followup=3)
## Expected number of events: 38
## Probability of event: 0.329
## Expected number of patients: 116

One Sample Likelihood-Ratio Based Approach

Reference

Introduction

For fixed \(d, V=\sum t_i \sim \operatorname{Gamma}(d, \lambda)\) and it is known \({ }^2\) that \[ W=\frac{2 d \lambda}{\hat{\lambda}} \sim \chi_{2 d}^2 \] although this result is approximate for general censoring patterns. Under \(H_0: \lambda=\lambda_0\), we need to find a constant \(k\) such that \(\alpha=P\left(1 / \hat{\lambda}>k \mid \lambda=\lambda_0\right)=P\left(W>2 d k \lambda_0\right)\). Thus we have \(\chi_{2 d, \alpha}^2=2 d k \lambda_0\) and hence \(k=\frac{\chi_{2 d_d \alpha}^2}{2 d \lambda_0}\). The power of the test is given by \[ 1-\beta=P\left(1 / \lambda>k \mid \hat{\lambda}=\lambda_A=P\left(W>2 d k \lambda_A\right)\right) . \] We have \(\chi_{2 d, 1-\beta}^2=2 d k \lambda_A \Rightarrow \chi_{2 d, 1-\beta}^2=\frac{\chi_{2 d, \alpha}^2 \lambda_A}{\lambda_0}\), hence \(\Delta=\frac{\lambda_0}{\lambda_A}=\frac{\chi_{2 d, \alpha}^2}{\chi_{2 d, 1-\beta}^2}\). For specified \(\alpha\), power \(1-\beta\), and ratio \(\Delta\), we may solve this for the required number of deaths, \(d\).

\(\Delta\) can be computed using the following function:

expLikeRatio = function(d, alpha, pwr){
  num = qchisq(alpha, df=(2*d), lower.tail=F)
  denom = qchisq(pwr, df=(2*d), lower.tail=F)
  Delta = num/denom
  Delta
}

To get the number of deaths \(d\) for a specified \(\Delta\), we define a new function \(L R(d)=\frac{\chi_{2 d, \alpha}^2}{\chi_{2 d, 1-\beta}^2}-\Delta\). The solution for \(L R(d)=0\) is the required number of deaths and is computed as:

expLRdeaths = function(Delta, alpha, pwr){
  LR = function(d, alpha, pwr, Delta){
    expLikeRatio(d, alpha, pwr) - Delta
  }
  # Find the root for the function LR(d)
  result = uniroot(f = LR, lower = 1, upper = 1000,
                   alpha = alpha, pwr = pwr, Delta = Delta)
  result$root
}

Calculation in R

Suppose that we are designing a Phase II oncology trial where we plan a 5% level (one-sided) test, and we need 80% power to detect a hazard ratio of 1.5. We can find the required number of deaths as follows:

ssc.onesample.LR(HR = 1.5, sig.level = 0.05, power = 0.8)
## Expected number of events: 37

One Sample Non-Parametric

Introduction

  1. The one-sample log-rank test, first proposed by Breslow (1975), allows for the comparison of the survival curve of a new treatment arm with that of a historical control.
  2. Non Parametric method calculate either estimates of accrual or power for null and alternative survival functions based on either design specifications of survival probability or median survival.
  3. The test statistic for survival probability is assumed to be based on the non-parametric estimate of the survival distribution. For median survival, a Brookmeyer-Crowley like test assumed.
  4. Ref

Reference

  • Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series.